I tried the first one on GPT-3.5-turbo and it got the answer wrong haha. Goes to show you how much better GPT-4 is...
GPT-3.5-TURBO:
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True.
A pure state of n qubits is described by an n-dimensional complex unit vector, also known as a quantum state vector. In Dirac notation, a pure state of n qubits can be represented as:
|ψ⟩ = a_0|0⟩ + a_1|1⟩ + ... + a_(2^n-1)|2^n-1⟩
where a_i are complex numbers and |i⟩ represents the i-th computational basis state. The normalization condition |a_0|^2 + |a_1|^2 + ... + |a_(2^n-1)|^2 = 1 must also be satisfied.
(hallucination?)
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GPT-4:
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False. A pure state of n qubits is described by a 2^n-dimensional complex unit vector. In a quantum system, each qubit can be in a superposition of two basis states, so for n qubits, there are 2^n possible basis states in the overall Hilbert space.
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Interesting to note that with GPT-3.5-turbo, a single followup chain-of-reasoning prompt ("Let's think step by step", i.e. no introduction of a priming prompt like "Are you sure?" or "Why are you wrong?") helps the model identify the correct answer:
GPT-3.5-TURBO:
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Sure, let's break it down step by step:
Qubits: A qubit is the fundamental unit of quantum information. It is a quantum analogue to the classical bit. While a classical bit can take one of two values, 0 or 1, a qubit can be in a superposition of both 0 and 1 states simultaneously.
Basis states: For a single qubit, there are two basis states, often denoted as |0> and |1>. A qubit can be in any linear combination (superposition) of these basis states, represented as α|0> + β|1>, where α and β are complex numbers, and the sum of the magnitudes squared (|α|^2 + |β|^2) equals 1.
Multiple qubits: When you have a system of n qubits, each qubit can be in a superposition of its basis states. The overall state of the n-qubit system is described by the tensor product of the individual qubit states. The number of basis states in the combined system is equal to the product of the number of basis states of each individual qubit.
n-dimensional vector: For n classical bits, you would indeed need an n-dimensional vector to describe their state. However, with n qubits in a quantum system, the number of basis states grows exponentially, as opposed to linearly in the classical case.
2^n-dimensional complex unit vector: As we mentioned earlier, each qubit can be in a superposition of two basis states. Therefore, for n qubits, there are 2^n possible basis states. The overall state of an n-qubit system is represented by a 2^n-dimensional complex unit vector, with each component corresponding to the probability amplitude of one of the 2^n basis states.
In summary, a pure state of n qubits is described by a 2^n-dimensional complex unit vector, not an n-dimensional one.
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